Units and dimensions: What is the SI unit of dynamic viscosity (μ) used in bioprocess calculations?

Introduction:
Keeping units straight is essential for Reynolds number, pressure drop, and mass-transfer calculations. Dynamic viscosity has a specific SI unit that also ties directly to the Pascal-second (Pa·s).

Given Data / Assumptions:

• We focus on dynamic viscosity μ (not kinematic viscosity ν).
• SI base units are used consistently.
• Newton’s law of viscosity: shear stress τ = μ * (du/dy).

Concept / Approach:
Shear stress τ has units of Pascals (Pa = N·m^-2 = kg·m^-1·s^-2). Shear rate du/dy has units s^-1. Rearranging τ = μ * (du/dy) gives μ = τ / (du/dy), so μ has units (kg·m^-1·s^-2) / s^-1 = kg·m^-1·s^-1, which is identical to Pa·s.

Step-by-Step Solution:
Start with τ = μ * γ̇ (where γ̇ is shear rate).Use [τ] = Pa = kg·m^-1·s^-2.Use [γ̇] = s^-1.Compute [μ] = [τ]/[γ̇] = (kg·m^-1·s^-2)/(s^-1) = kg·m^-1·s^-1 = Pa·s.

Verification / Alternative check:
Common reference values: water at 20°C has μ ≈ 1.0×10^-3 kg·m^-1·s^-1 (1 mPa·s), consistent with Pa·s units.

Why Other Options Are Wrong:
kg·m^-2·s^-2 or kg·m^-3·s^-1 or kg·m^-1·s^-2: these correspond to other derived quantities (e.g., pressure, density flow terms, or stress), not dynamic viscosity.

Common Pitfalls:

• Confusing dynamic viscosity μ with kinematic viscosity ν = μ/ρ, which has units m^2·s^-1.
• Dropping or misplacing exponents, leading to dimensional inconsistency in calculations.

Final Answer:
kg·m^-1·s^-1 (equivalent to Pa·s)